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We are to list the unknowns and the equations for a given flow situation.

Analysis

There are only three unknowns in this problem,

u

,

v

, and

P

(or

P

′

)

. Thus, we require three equations:

continuity

,

x

momentum

(or

x

component of Navier-Stokes), and

y

momentum

(or

y

component of Navier-Stokes). Theseequations, when combined with the appropriate boundary conditions, are sufficient to solve the problem.

Discussion

The actual equations to be solved by the computer are discretized versions of the differential equations.

15-2C

Solution

We are to define several terms or phrases and provide examples.

Analysis

(

a

)

A computational domain is a region in space (either 2-D or 3-D) in which the numerical equations of fluid floware solved by CFD

. The computational domain is bounded by edges (2-D) or faces (3-D) on which boundaryconditions are applied.(

b

)

A mesh is generated by dividing the computational domain into tiny cells

. The numerical equations are then solvedin each cell of the mesh. A mesh is also called a grid.(

c

)

A transport equation is a differential equation representing how some property is transported through a flowfield

. The transport equations of fluid mechanics are conservation equations. For example, the continuity equation is adifferential equation representing the transport of mass, and also conservation of mass. The Navier-Stokes equation is adifferential equation representing the transport of linear momentum, and also conservation of linear momentum.(

d

)

Equations are said to be coupled when at least one of the variables (unknowns) appears in more than oneequation

. In other words, the equations cannot be solved alone, but must be solved simultaneously with each other.This is the case with fluid mechanics since each component of velocity, for example, appears in the continuity equationand in all three components of the Navier-Stokes equation.

Discussion

Students’ definitions should be in their own words.

15-3C

Solution

We are to discuss the difference between nodes and intervals and analyze a given computational domain interms of nodes and intervals.

Analysis

Nodes are points along an edge of a computational domain that represent the vertices of cells

. In other words, they are the points where corners of the cells meet.

Intervals, on the other hand, are short line segments betweennodes

. Intervals represent the small edges of cells themselves. In Fig. P15-3C there are

For a given computational domain with specified nodes and intervals, we are to compare a structured gridand an unstructured grid and discuss.

Analysis

We construct the two grids in the figure: (a) structured, and (b) unstructured.(a) (b)There are 5

×

4 = 20 cells in the structured grid, and there are 36 cells in the unstructured grid.

Discussion

Depending on how individual students construct their unstructured grid, the shape, size, and number of cellsmay differ considerably.

15-5C

Solution

For a given computational domain with specified nodes and intervals, we are to compare a structured meshand a polyhedral mesh and discuss.

Analysis

We construct the two grids in the figure: (a) structured, and (b) unstructured polyhedral. We show two other options in (c) and (d). There are many possible answers for the polyhedral mesh, depending on how large you want your cells to be.(a) (b)(c) (d)There are 5

×

4 = 20 cells in the structured grid. There are 22 cells in polyhedral grid (b). There are some cells with 3 sides,4 sides, and 5 sides, as required. Compared to the triangular mesh with 36 cells, we have reduced the cell countconsiderably. In (c) and (d), there are 21 cells and 18 cells respectively. In case (d) we have reduced the cell count belowthat of even the structured grid. In that case, 3 of the cells have 6 sides each. The cell reduction is particularly useful inlarge 3-D problems where CPU time and computer memory are important limitations.

Discussion

Note that the node distribution along the boundaries is identical in each case, but we have great flexibility inhow we create the grid. Depending on how individual students construct their unstructured grid, the shape, size, andnumber of cells may differ considerably.